k=  7 m= 3: deg phi_(m+1) = 2, (C1) True, (C2) True, (C3) True  -> OK
k=  9 m= 4: deg phi_(m+1) = 2, (C1) True, (C2) True, (C3) True  -> OK
k= 11 m= 5: deg phi_(m+1) = 3, (C1) True, (C2) True, (C3) True  -> OK
k= 13 m= 6: deg phi_(m+1) = 3, (C1) True, (C2) True, (C3) True  -> OK
k= 15 m= 7: deg phi_(m+1) = 4, (C1) True, (C2) True, (C3) True  -> OK
k= 17 m= 8: deg phi_(m+1) = 4, (C1) True, (C2) True, (C3) True  -> OK
k= 19 m= 9: deg phi_(m+1) = 5, (C1) True, (C2) True, (C3) True  -> OK
k= 21 m=10: deg phi_(m+1) = 5, (C1) True, (C2) True, (C3) True  -> OK
k= 23 m=11: deg phi_(m+1) = 6, (C1) True, (C2) True, (C3) True  -> OK
k= 25 m=12: deg phi_(m+1) = 6, (C1) True, (C2) True, (C3) True  -> OK
k= 27 m=13: deg phi_(m+1) = 7, (C1) True, (C2) True, (C3) True  -> OK
k= 29 m=14: deg phi_(m+1) = 7, (C1) True, (C2) True, (C3) True  -> OK
k= 31 m=15: deg phi_(m+1) = 8, (C1) True, (C2) True, (C3) True  -> OK
PART 1: all conditions hold for odd k in [7, 31]
PART 2 (exact expansion for k = 7, 9):
  k=7: coefficient of t^1 in p q^6 P (minus k beta p (q^6 - 1)) reduces to 0
  k=7: coefficient of t^4 in p q^6 P  reduces to 0
  k=7: coefficient of t^7 in p q^6 P  reduces to 0
  k=7: coefficient of t^10 in p q^6 P  reduces to 0
  k=7: coefficient of t^13 in p q^6 P  reduces to 0
  k=7: numeric check of t = (1/(m sigma)) sum lam^-j P(lam^j t) at three points: max error 1.59e-56; |sigma| = 13.0499
  k=7: EXACT IDENTITY VERIFIED
  k=9: coefficient of t^1 in p q^8 P (minus k beta p (q^8 - 1)) reduces to 0
  k=9: coefficient of t^5 in p q^8 P  reduces to 0
  k=9: coefficient of t^9 in p q^8 P  reduces to 0
  k=9: coefficient of t^13 in p q^8 P  reduces to 0
  k=9: coefficient of t^17 in p q^8 P  reduces to 0
  k=9: numeric check of t = (1/(m sigma)) sum lam^-j P(lam^j t) at three points: max error 2.26e-57; |sigma| = 1318.16
  k=9: EXACT IDENTITY VERIFIED
RESULT: PASS
